# The Term Structure of Interest Rates. The relationship between yield to maturity and maturity. Information on expected future short term rates (short.

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The Term Structure of Interest Rates

The relationship between yield to maturity and maturity. Information on expected future short term rates (short rate) might be implied from the yield curve. (short rate vs. spot (average) rate) The yield curve is a graph that displays the relationship between yield and maturity. Three major theories are proposed to explain the observed yield curve. Overview of Term Structure of Interest Rates

Yields Maturity Upward Sloping Downward Sloping Flat Spot Rates on Yield Curves

Expected Interest Rates in Coming Years Expected One-Year Short Rates in Coming Years YearInterest Rate 0 (today) 8% 110% 211% 311%

Pricing of Zero Coupon Bonds using Expected Rate PV n = Present Value of \$1 in n periods r 1 = One-year short rate for period 1 r 2 = One-year short rate for period 2 r n = One-year short rate for period n y n= yield for a zero coupon security with a maturity of n

 Thus Short rates  Spot rates  But what about the other way around?

f n = one-year forward rate for period n y n = yield for a security with a maturity of n Forward Rates from Observed Long-Term Rates

Example of Forward Rates 4 yr = 9.993%3yr = 9.660%fn = ? (1.0993) 4 = (1.0966) 3 (1+f n ) (1.46373) / (1.31870) = (1+f n ) f n =.10998 or 11% Note: this is expected rate that was used in the prior example.

Downward Sloping Spot Yield Curve Zero-Coupon RatesBond Maturity 12%1 11.75%2 11.25%3 10.00%4 9.25%5

Forward Rates for Downward Sloping Yield Curve 1yr Forward Rates 1yr[(1.1175) 2 / 1.12] - 1 =0.115006 2yrs[(1.1125) 3 / (1.1175) 2 ] - 1 =0.102567 3yrs[(1.1) 4 / (1.1125) 3 ] - 1 =0.063336 4yrs[(1.0925) 5 / (1.1) 4 ] - 1 =0.063008

Expectations Liquidity Preference -Upward bias over expectations Market Segmentation -Preferred Habitat Theories of Term Structure

Expectations Theory Observed long-term rate is a function of today’s short-term rate and expected future short-term rates. Long-term and short-term securities are perfect substitutes. Forward rates that are calculated from the yield on long-term securities are market consensus expected future short-term rates.

Long-term bonds are more risky. Investors will demand a premium for the risk associated with long-term bonds. The yield curve has an upward bias built into the long-term rates because of the risk premium. Forward rates contain a liquidity premium and are not equal to expected future short-term rates. Liquidity Premium Theory

Liquidity Premiums and Yield Curves Yields Maturity Liquidity Premium Short Rates Observed Yield Curve

Liquidity Premiums and Yield Curves Yields Maturity Liquidity Premium Short Rates Observed Yield Curve

Short- and long-term bonds are traded in distinct markets. Trading in the distinct segments determines the various rates. Observed rates are not directly influenced by expectations. Preferred Habitat: -Modification of market segmentation -Investors will switch out of preferred maturity segments if premiums are adequate. Market Segmentation and Preferred Habitat

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